






from __future__ import annotations

from fractions import Fraction

def        UpperCAmelCase__		(						lowerCamelCase_       :       int   ,  lowerCamelCase_       :       int				):
			return (
			    num != den and num % 1_0 == den // 1_0 and (num // 1_0) / (den % 1_0) == num / den
			)

def        UpperCAmelCase__		(						lowerCamelCase_       :       int				):
			__a						:  int									=     []
			__a						:  int									=     1_1
			__a						:  List[Any]									=     int('1' + '0' * digit_len				)
			for num in range(lowerCamelCase_   ,  lowerCamelCase_				):
						while den <= 9_9:
									if (num != den) and (num % 1_0 == den // 1_0) and (den % 1_0 != 0):
												if is_digit_cancelling(lowerCamelCase_   ,  lowerCamelCase_				):
															solutions.append(f'''{num}/{den}'''				)
									den += 1
						num += 1
						__a						:  Union[str, Any]									=     1_0
			return solutions

def        UpperCAmelCase__		(						lowerCamelCase_       :       int = 2				):
			__a						:  Optional[Any]									=     1.0
			for fraction in fraction_list(lowerCamelCase_				):
						__a						:  int									=     Fraction(lowerCamelCase_				)
						result *= frac.denominator / frac.numerator
			return int(lowerCamelCase_				)


if __name__ == "__main__":
							print(solution())











"""simple docstring"""




def        a__					(							snake_case__ = 50_00_00_00							)		->   int:
			lowerCamelCase      =   set()
			lowerCamelCase      =   int((limit - 24) ** (1 / 2)							)

			lowerCamelCase      =   set(range(3			,					prime_square_limit + 1			,					2							)							)
			primes.add(2							)
			for p in range(3			,					prime_square_limit + 1			,					2							):
						if p not in primes:
									continue
						primes.difference_update(set(range(p * p			,					prime_square_limit + 1			,					snake_case__							)							)							)

			for primea in primes:
						lowerCamelCase      =   primea * primea
						for primea in primes:
									lowerCamelCase      =   primea * primea * primea
									if square + cube >= limit - 16:
												break
									for primea in primes:
												lowerCamelCase      =   primea * primea * primea * primea
												lowerCamelCase      =   square + cube + tetr
												if total >= limit:
															break
												ret.add(snake_case__							)

			return len(snake_case__							)


if __name__ == "__main__":
					print(F"""{solution() = }""")



